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I have a function $f\in \mathcal{S}$ (i.e of Schwartz class), and I want to show there exist constants $C,k>0$ s.t $$\|f\|_p \leq C(\sup_{x\in \mathbb{R}} |f(x)| + \sup_{\mathbb{x\in \mathbb{R}}} |x^k f(x)|)$$ for every $ p \in [1,\infty]$.

For $p=1,\infty$ it's obvious from definition, I mean I can take f with compact support and this will prove for $p=1$, for $p=\infty$ it's trivial.

But for $ p \in (1,\infty)$ I find myself at a mess, I need to do integration by parts inductively but I don't seem to find the right approach, I guess I need to use here Leibnitz general product rule, but I don't see how to come to suitable constants.

Any help , is appreciated.

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2 Answers 2

up vote 1 down vote accepted

Hint: $$ \|f\|_p^p=\int_{\mathbb R}\frac{|f(x)|^p(1+x^2)}{1+x^2}\,dx. $$

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We have $$ \|f\|_\infty=\sup_{x\in\mathbb{R}}|f(x)| $$ and $$ \|f\|_1\le\pi\sup_{x\in\mathbb{R}}|f(x)|(1+x^2) $$ since $\int_{\mathbb{R}}\frac{\mathrm{d}x}{1+x^2}=\pi$.

Now apply inequality $(3)$ with $p=1$.

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